How one would compute cos(5). 

As always, there are many ways to skin the cat, but here is one simple approach:

Step 1: set y = [2Pi - 5], then cos(5) = cos(2Pi - y) = cos(y).

Now y ~ 1.2 > 1, so still not too good for straight Taylor series (around 0).

Step 2: cos(y) = sin(Pi/2 - y). Note that Pi/2 -y ~ 0.29 < 1 and small enough
for the Taylor Series to converge fast. 

Step 3: Now you can expand sin(Pi/2 -y) = (Pi/2 - y) - 1/6*(Pi/2 -y)^3 ....

As an alternative, you can consider "shifting" the argument by 3/2*Pi